Here we study theoretically the effects of anisotropy on the thermoelectric performance of p-type AgBiSe2. We present an apparent realization of the thermoelectric benefits of one-dimensional plate-like carrier pocket anisotropy in the valence band of this material. Based on first principles calculations we find a substantial anisotropy in the electronic structure, likely favorable for thermoelectric performance, in the valence bands of the hexagonal phase of the silver chalcogenide thermoelectric AgBiSe2, while the conduction bands are more isotropic, and in our experiments do not attain high performance. AgBiSe2 has already exhibited a ZT value of 1.5 in a high-temperature disordered fcc phase, but room-temperature performance has not been demonstrated. We develop a theory for the ability of anisotropy to decouple the density-of-states and conductivity effective masses, pointing out the influence of this effect in the high performance thermoelectrics Bi2Te3 and PbTe. From our first principles and Boltzmann transport calculations we find that p-type AgBiSe2 has substantial promise as a room temperature thermoelectric, and estimate its performance.